Geometric and Functional Analysis

, Volume 20, Issue 1, pp 53–67

Ergodic Subequivalence Relations Induced by a Bernoulli Action

Article

Abstract

Let Γ be a countable group and denote by \({\mathcal{S}}\) the equivalence relation induced by the Bernoulli action \({\Gamma\curvearrowright [0, 1]^{\Gamma}}\), where [0, 1]Γ is endowed with the product Lebesgue measure. We prove that, for any subequivalence relation \({\mathcal{R}}\) of \({\mathcal{S}}\), there exists a partition {Xi}i≥0 of [0, 1]Γ into \({\mathcal{R}}\)-invariant measurable sets such that \({\mathcal{R}_{\vert X_{0}}}\) is hyperfinite and \({\mathcal{R}_{\vert X_{i}}}\) is strongly ergodic (hence ergodic and non-hyperfinite), for every i ≥ 1.

Keywords and phrases

Bernoulli action deformation/rigidity ergodic subequivalence relation malleable strongly ergodic 

2010 Mathematics Subject Classification

37A20 37A15 

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Copyright information

© Birkhäuser / Springer Basel AG 2010

Authors and Affiliations

  1. 1.Math. DeptUCLALos AngelesUSA
  2. 2.IMARBucharestRomania

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